Binomial Theorem — MATHS STD 11 Science — Question

1. $\frac{405}{256}$

Solution:

Suppose x⁴ occurs at the (r + 1)^th term in the given expansion.

Then, we have

$\text{T}_{\text{r}+1}={^\text{10}}\text{C}_{\text{r}}\Big(\frac{\text{x}}{2}\Big)^{10-\text{r}}\Big(\frac{-3}{2\text{x}^{2}}\Big)$

$=(-1)^{\text{r}}\ {^\text{10}}\text{C}_{\text{r}}\frac{3^{\text{r}}}{2^{10-\text{r}}}\text{x}^{10-\text{r}-2\text{r}}$

For this term to contain x⁴, we must have

$10-3\text{r}=4$

$\Rightarrow \text{r}=2$

$\therefore$ Required coefficient $={^\text{10}}\text{C}_{\text{2}}\frac{3^{2}}{2^{8}}=\frac{10\times9\times9}{2\times2^{8}}=\frac{405}{256}$